# Sequences and Series: Modelling

## Sequences and Series: Modelling

## Understanding Modelling of Sequences and Series

**Modelling**involves using a mathematical structure to represent real-world phenomena and then to predict or explain the phenomena.- In
**sequences and series**, the modelling process may involve setting up a sequence to describe a situation, or fitting a sequence to data. - A
**series**is the summed terms of a sequence, whereas the sequence is simply the list of numbers.

## Arithmetic Sequences and Series

- An
**arithmetic sequence**is a sequence of numbers in which the difference of between any two successive members is constant. - The nth term of an arithmetic sequence can be given by
**a + (n-1)d**, where a is the first term and d is the common difference. - The sum of an arithmetic series,
`S_n`

, of n terms can be calculated using the formula**S_n = n/2(2a + (n - 1)d)**.

## Geometric Sequences and Series

- A
**geometric sequence**is a sequence of numbers in which the ratio of any two successive members is constant. This constant is often called the common ratio. - The nth term of a geometric sequence can be given by
**ar^(n-1)**, where a is the first term and r is the common ratio. -
The sum of a geometric series, `S_n`

, of n terms can be calculated using the formula**S_n = a(1 - r^n) / (1 - r)**, wherer < 1.

## Using Sequences and Series in Modelling

- In modelling physical or biological phenomena,
**arithmetic and geometric sequences and series**are often used to represent or approximate the phenomena. - The choice between arithmetic and geometric sequences and series often depends on the nature of the phenomena, such as whether the phenomena exhibit constant change or exponential change.

## The Limit of a Series

- The
**limit**of a series is a value that the series approaches as the number of terms goes to infinity. -
For a geometric series where r < 1, the series has a limit, given by **a / (1 - r)**.

## Fibonacci and Other Sequences

- There are many sequences other than arithmetic and geometric sequences, such as the
**Fibonacci sequence**, which can also be used in modelling. - Fibonacci sequence is defined by the recurrence relation
**f(n) = f(n - 1) + f(n - 2)**with initial conditions f(0) = 0 and f(1) = 1. - In a similar way to arithmetic and geometric sequences, the properties of these sequences can be analysed, and the sequences can be used to model real-world phenomena.